Once tractability of the finite-horizon underlying optimization problem is insured, it is possible to implement the resulting optimal solution using an MPC approach or an RHC approach. In the former case [1]}, the optimization problem is resolved at each step and only the first control input is implemented. In the latter case [2]}, the optimization problem is resolved every \(N\)  steps (with \(N\)  being the horizon length) and the entire sequence of \(N\)  input vectors is implemented. Both of these approaches are shown to provide stability under the assumption that the zero-input and zero-noise system is asymptotically stable, which translates into the condition that the state matrix \(A\)  is Schur stable. At a first glance, this assumption might seem restrictive. However, the problem of ensuring bounded variance of linear Gaussian systems with bounded control inputs is, to our knowledge, still open, and here we are considering the problem of controlling a linear system with bounded control input and possibly unbounded noise. It is known that for discrete-time systems without any noise acting on the system it is possible to achieve global stability if and only if the matrix \(A\)  is neutrally stable [3]}.
